Optimal. Leaf size=112 \[ \frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 b^4 d}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}} \]
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Rubi [A] time = 0.0896406, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {4045, 3769, 3771, 2641} \[ \frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^4 d}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 4045
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx &=\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{(5 A+7 C) \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx}{7 b^2}\\ &=\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{(5 A+7 C) \int \sqrt{b \sec (c+d x)} \, dx}{21 b^4}\\ &=\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{\left ((5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b^4}\\ &=\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^4 d}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.691509, size = 79, normalized size = 0.71 \[ \frac{\frac{4 (5 A+7 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{\sqrt{\cos (c+d x)}}+2 \sin (c+d x) (3 A \cos (2 (c+d x))+13 A+14 C)}{42 b^3 d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.265, size = 241, normalized size = 2.2 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}} \left ( 5\,iA\sin \left ( dx+c \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}+7\,iC\sin \left ( dx+c \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-3\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-5\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}-7\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5\,A\cos \left ( dx+c \right ) +7\,C\cos \left ( dx+c \right ) \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt{b \sec \left (d x + c\right )}}{b^{4} \sec \left (d x + c\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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